No, the natural numbers aren't a language structure. A languagestructure looks like this:

<a set U, family of relations of various arities on U, family of functions of various arities on U, distinguished elements of U>.

The details will depend on the language. U may be the set of naturalnumbers, indeed (for first order languages) it can always be chosen tobe N so long as it isn't finite. But that has got nothing to do withyour false claim that G\"odel's incompleteness theorem relates tolanguage structure.

-- When a true genius appears in the world, you may know him by this sign, that the dunces are all in confederacy against him.Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting